the eigenvalues are

, i need help finding the eigenvectors

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- Sep 12th 2009, 07:29 PMmancillaj3need help finding eigenvectors

the eigenvalues are

, i need help finding the eigenvectors - Sep 13th 2009, 03:03 AMgarymarkhov
I'm not sure why you've used the notation , but I'll assume that you just mean you have some matrix and you want to find the eigenvectors.

Anyway, you do so in the same way you always find the eigenvectors. Nothing changes. First, find the eigenvalues as you've done. Choose one of the eigenvalues, say, , and subtract it from the diagonal entries:

Solving for in will give us the eigenvectors just like it does when we have real eigenvalues. [**Added:**This is from the equation , which can be rewritten as ].

We know that this matrix is singular because*we made it singular*. We did that by subtracting just the right eigenvalues that would make the determinant zero! So Just set and solve for . In other words, we want to make sure this is true:

Super. That means we just need to solve for in

It turns out that so we can write and know it's true. Do the matrix multiplication to convince yourself.

So you've found one**basis**eigenvector, which is . There's one more**basis**for the other eigenvalue, but I think you can work that out yourself now :) - Sep 13th 2009, 03:56 AMHallsofIvy
I prefer a more basic method. The

**definition**of "eigenvalue" is that " is an eigenvalue of matrix A if and only if there exist a non-zero vector v, an**eigenvector**corresponding to eigenvalue , such that ."

Since is an eigenvalue, for any eigenvector we must have

That gives us two equations, and . The first equation is the same as . The second equation is the same as . Dividing both sides by (which is the same as multiplying by , we get exactly the same as the first equation!

That**had**to happen. There exist an infinite number of eigenvectors corresponding to a given eigenvalue. The problem**must**reduce to one equation in one unknown. Take x to be anything you like and solve for y. If we take x= 6 to get rid of the fraction, so an eigevector corresponding to eigenvalue is

and**any**eigevector corresponding to that eigenvalue is of the form

for some number, x.

Do the same thing to find an eigenvector corresponding to eigenvalue .